Number Systems – Class 11

The study of number systems forms the foundation for higher mathematics. In Class 11, students explore different types of numbers, their properties, and how they relate to each other. This chapter builds on your earlier knowledge and introduces new concepts essential for advanced math.

1. Types of Number Systems

• Natural Numbers (N): Counting numbers starting from 1 (1, 2, 3, ...).
• Whole Numbers (W): Natural numbers including 0 (0, 1, 2, 3, ...).
• Integers (Z): Positive and negative whole numbers, including 0 (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers (Q): Numbers that can be expressed as p/q, where p and q are integers and q ≠ 0.
• Irrational Numbers: Numbers that cannot be expressed as p/q (e.g., √2, π).
• Real Numbers (R): All rational and irrational numbers.

2. Representation of Real Numbers on the Number Line

• Every real number can be represented as a unique point on the number line.
• Decimals, fractions, and irrational numbers can be located using appropriate methods (e.g., successive magnification for irrationals).

3. Properties of Rational and Irrational Numbers

• Sum, difference, and product of two rational numbers is rational.
• Sum or product of a rational and an irrational number is irrational (except in special cases).
• Sum or product of two irrational numbers can be rational or irrational.

4. Decimal Expansions

• Rational numbers have either terminating or non-terminating repeating decimal expansions.
• Irrational numbers have non-terminating, non-repeating decimal expansions.

5. Laws of Exponents for Real Numbers

• a^m × aⁿ = a^m+n
• a^m ÷ aⁿ = a^m-n
• (a^m)ⁿ = a^mn
• (ab)ⁿ = aⁿbⁿ
• a⁰ = 1 (a ≠ 0)
• a^-n = 1/aⁿ

6. Important Concepts

• Surds: Irrational numbers with roots (e.g., √2, ³√5).
• Rationalization: Process of converting a denominator to a rational number.
• Absolute Value: The non-negative value of a number (|x|).

7. Example Problems

1. Classify the following as rational or irrational: 3/4, √5, 0.333..., π.
2. Express 0.142857142857... as a rational number.
3. Find the value of (2³ × 2^-5).
4. Rationalize the denominator: 1/(√3 + 1).
5. Represent √2 on the number line.

8. Practice Exercises

• Identify whether the following numbers are rational or irrational: 0, -7, √7, 22/7, 1.414213...
• Write the decimal expansion of 5/8 and state whether it is terminating or non-terminating.
• Simplify: (3² × 3^-5) / 3^-2
• Rationalize: 1/(2 - √3)
• Find the absolute value of -15.

9. Summary

• Number systems include natural, whole, integers, rational, irrational, and real numbers.
• Every real number can be represented on the number line.
• Decimal expansions help distinguish between rational and irrational numbers.
• Laws of exponents are essential for simplifying expressions.